Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
ACTIVE(U11(tt, N, XS)) → U12¹(tt, N, XS)
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
ACTIVE(U12(tt, N, XS)) → SND(splitAt(N, XS))
ACTIVE(U12(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
ACTIVE(U21(tt, X)) → U22¹(tt, X)
ACTIVE(U22(tt, X)) → MARK(X)
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
ACTIVE(U31(tt, N)) → U32¹(tt, N)
ACTIVE(U32(tt, N)) → MARK(N)
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
ACTIVE(U41(tt, N, XS)) → U42¹(tt, N, XS)
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
ACTIVE(U42(tt, N, XS)) → HEAD(afterNth(N, XS))
ACTIVE(U42(tt, N, XS)) → AFTERNTH(N, XS)
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
ACTIVE(U51(tt, Y)) → U52¹(tt, Y)
ACTIVE(U52(tt, Y)) → MARK(Y)
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
ACTIVE(U61(tt, N, X, XS)) → U62¹(tt, N, X, XS)
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
ACTIVE(U62(tt, N, X, XS)) → U63¹(tt, N, X, XS)
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
ACTIVE(U63(tt, N, X, XS)) → U64¹(splitAt(N, XS), X)
ACTIVE(U63(tt, N, X, XS)) → SPLITAT(N, XS)
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
ACTIVE(U64(pair(YS, ZS), X)) → PAIR(cons(X, YS), ZS)
ACTIVE(U64(pair(YS, ZS), X)) → CONS(X, YS)
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
ACTIVE(U71(tt, XS)) → U72¹(tt, XS)
ACTIVE(U72(tt, XS)) → MARK(XS)
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
ACTIVE(U81(tt, N, XS)) → U82¹(tt, N, XS)
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
ACTIVE(U82(tt, N, XS)) → FST(splitAt(N, XS))
ACTIVE(U82(tt, N, XS)) → SPLITAT(N, XS)
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
ACTIVE(afterNth(N, XS)) → U11¹(tt, N, XS)
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
ACTIVE(fst(pair(X, Y))) → U21¹(tt, X)
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
ACTIVE(head(cons(N, XS))) → U31¹(tt, N)
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
ACTIVE(natsFrom(N)) → CONS(N, natsFrom(s(N)))
ACTIVE(natsFrom(N)) → NATSFROM(s(N))
ACTIVE(natsFrom(N)) → S(N)
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
ACTIVE(sel(N, XS)) → U41¹(tt, N, XS)
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
ACTIVE(snd(pair(X, Y))) → U51¹(tt, Y)
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
ACTIVE(splitAt(0, XS)) → PAIR(nil, XS)
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
ACTIVE(splitAt(s(N), cons(X, XS))) → U61¹(tt, N, X, XS)
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
ACTIVE(tail(cons(N, XS))) → U71¹(tt, XS)
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
ACTIVE(take(N, XS)) → U81¹(tt, N, XS)
MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
MARK(U11(X1, X2, X3)) → U11¹(mark(X1), X2, X3)
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
MARK(U12(X1, X2, X3)) → U12¹(mark(X1), X2, X3)
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
MARK(snd(X)) → SND(mark(X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
MARK(splitAt(X1, X2)) → SPLITAT(mark(X1), mark(X2))
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
MARK(U42(X1, X2, X3)) → U42¹(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(head(X)) → HEAD(mark(X))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
MARK(afterNth(X1, X2)) → AFTERNTH(mark(X1), mark(X2))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U52(X1, X2)) → U52¹(mark(X1), X2)
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
MARK(U61(X1, X2, X3, X4)) → U61¹(mark(X1), X2, X3, X4)
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
MARK(U62(X1, X2, X3, X4)) → U62¹(mark(X1), X2, X3, X4)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
MARK(U63(X1, X2, X3, X4)) → U63¹(mark(X1), X2, X3, X4)
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
MARK(U64(X1, X2)) → U64¹(mark(X1), X2)
MARK(U64(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
MARK(pair(X1, X2)) → PAIR(mark(X1), mark(X2))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
MARK(U71(X1, X2)) → U71¹(mark(X1), X2)
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
MARK(U72(X1, X2)) → U72¹(mark(X1), X2)
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
MARK(U81(X1, X2, X3)) → U81¹(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
MARK(U82(X1, X2, X3)) → U82¹(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
MARK(fst(X)) → FST(mark(X))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
MARK(natsFrom(X)) → NATSFROM(mark(X))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
MARK(nil) → ACTIVE(nil)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → TAIL(mark(X))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)
SND(mark(X)) → SND(X)
SND(active(X)) → SND(X)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)
U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(X1, mark(X2)) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)
FST(mark(X)) → FST(X)
FST(active(X)) → FST(X)
NATSFROM(mark(X)) → NATSFROM(X)
NATSFROM(active(X)) → NATSFROM(X)
S(mark(X)) → S(X)
S(active(X)) → S(X)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
TAIL(mark(X)) → TAIL(X)
TAIL(active(X)) → TAIL(X)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 31 SCCs with 64 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(X1, mark(X2)) → TAKE(X1, X2)
TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, mark(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(mark(X1), X2) → TAKE(X1, X2)
TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(mark(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  TAKE(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > TAKE₂

Status:

TAKE₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(active(X1), X2) → TAKE(X1, X2)
TAKE(X1, active(X2)) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(X1, active(X2)) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAKE(active(X1), X2) → TAKE(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAKE(active(X1), X2) → TAKE(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAKE(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(mark(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1)  =  TAIL(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > TAIL₁

Status:

mark₁: [1]
TAIL₁: [1]

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TAIL(active(X)) → TAIL(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TAIL(active(X)) → TAIL(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TAIL(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) TRUE

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, mark(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(mark(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  SEL(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SEL₂

Status:

mark₁: [1]
SEL₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(active(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(X1, active(X2)) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(29) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SEL(active(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(active(X1), X2) → SEL(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(32) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(33) TRUE

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > S₁

Status:

mark₁: [1]
S₁: [1]

The following usable rules [FROCOS05] were oriented: none

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(39) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(40) TRUE

(41) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NATSFROM(active(X)) → NATSFROM(X)
NATSFROM(mark(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(mark(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NATSFROM(x1)  =  NATSFROM(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > NATSFROM₁

Status:

mark₁: [1]
NATSFROM₁: [1]

The following usable rules [FROCOS05] were oriented: none

(43) Obligation:

Q DP problem:
The TRS P consists of the following rules:

NATSFROM(active(X)) → NATSFROM(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NATSFROM(active(X)) → NATSFROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NATSFROM(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(46) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(47) TRUE

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FST(active(X)) → FST(X)
FST(mark(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(mark(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FST(x1)  =  FST(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > FST₁

Status:

FST₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

FST(active(X)) → FST(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FST(active(X)) → FST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FST(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(53) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(54) TRUE

(55) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U82^1₂

Status:

active₁: [1]
U82^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(63) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(64) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U82^1₂

Status:

mark₁: [1]
U82^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(65) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(66) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(67) TRUE

(68) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U81^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(70) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U81^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(72) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U81^1₂

Status:

active₁: [1]
U81^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(76) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U81^1₂

Status:

U81^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(78) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(79) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(80) TRUE

(81) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U72¹(X1, mark(X2)) → U72¹(X1, X2)
U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(82) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, mark(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(83) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U72¹(mark(X1), X2) → U72¹(X1, X2)
U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(84) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(mark(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  U72¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U72^1₂

Status:

mark₁: [1]
U72^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(85) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U72¹(active(X1), X2) → U72¹(X1, X2)
U72¹(X1, active(X2)) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(X1, active(X2)) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U72¹(active(X1), X2) → U72¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U72¹(active(X1), X2) → U72¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U72¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(90) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(91) TRUE

(92) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U71¹(X1, mark(X2)) → U71¹(X1, X2)
U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, mark(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U71¹(mark(X1), X2) → U71¹(X1, X2)
U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(mark(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  U71¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U71^1₂

Status:

U71^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U71¹(active(X1), X2) → U71¹(X1, X2)
U71¹(X1, active(X2)) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(X1, active(X2)) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(98) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U71¹(active(X1), X2) → U71¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X1), X2) → U71¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(101) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(102) TRUE

(103) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > CONS₂

Status:

CONS₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(active(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(110) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(111) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(112) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(113) TRUE

(114) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(X1, mark(X2)) → PAIR(X1, X2)
PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, mark(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(mark(X1), X2) → PAIR(X1, X2)
PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(mark(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  PAIR(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > PAIR₂

Status:

PAIR₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(118) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(active(X1), X2) → PAIR(X1, X2)
PAIR(X1, active(X2)) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(119) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(X1, active(X2)) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(120) Obligation:

Q DP problem:
The TRS P consists of the following rules:

PAIR(active(X1), X2) → PAIR(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PAIR(active(X1), X2) → PAIR(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PAIR(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(123) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(124) TRUE

(125) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U64¹(X1, mark(X2)) → U64¹(X1, X2)
U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(126) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(X1, mark(X2)) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(127) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U64¹(mark(X1), X2) → U64¹(X1, X2)
U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(128) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(mark(X1), X2) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  U64¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U64^1₂

Status:

U64^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(129) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U64¹(active(X1), X2) → U64¹(X1, X2)
U64¹(X1, active(X2)) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(130) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(X1, active(X2)) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(131) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U64¹(active(X1), X2) → U64¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(132) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U64¹(active(X1), X2) → U64¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U64¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(133) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(134) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(135) TRUE

(136) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(137) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, active(X3), X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U63^1₁

Status:

active₁: [1]
U63^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(138) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(139) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, mark(X3), X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U63^1₁

Status:

U63^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(140) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, X3, mark(X4)) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U63^1₁

Status:

U63^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(142) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, X2, X3, active(X4)) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U63^1₁

Status:

active₁: [1]
U63^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, mark(X2), X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(mark(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x1, x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U63^1₃

Status:

U63^1₃: [1,3,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(146) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)
U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(147) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(X1, active(X2), X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(148) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(149) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U63¹(active(X1), X2, X3, X4) → U63¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U63¹(x1, x2, x3, x4)  =  U63¹(x1, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U63^1₂

Status:

active₁: [1]
U63^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(150) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(151) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(152) TRUE

(153) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(154) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, active(X3), X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U62^1₁

Status:

active₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(155) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(156) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, mark(X3), X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U62^1₁

Status:

mark₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(157) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(158) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, X3, mark(X4)) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U62^1₁

Status:

mark₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(159) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(160) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, X2, X3, active(X4)) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U62^1₁

Status:

active₁: [1]
U62^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(161) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(162) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, mark(X2), X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(mark(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x1, x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U62^1₃

Status:

mark₁: [1]
U62^1₃: [1,3,2]

The following usable rules [FROCOS05] were oriented: none

(163) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)
U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(X1, active(X2), X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(166) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(active(X1), X2, X3, X4) → U62¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4)  =  U62¹(x1, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U62^1₂

Status:

active₁: [1]
U62^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(167) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(168) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(169) TRUE

(170) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, active(X3), X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U61^1₁

Status:

active₁: [1]
U61^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(173) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, mark(X3), X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U61^1₁

Status:

U61^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(174) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(175) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, X3, mark(X4)) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U61^1₁

Status:

U61^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(176) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(177) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, X2, X3, active(X4)) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x4)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U61^1₁

Status:

active₁: [1]
U61^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(178) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, mark(X2), X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(mark(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x1, x2, x4)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U61^1₃

Status:

U61^1₃: [1,3,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(180) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)
U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(181) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(X1, active(X2), X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(182) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(183) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X1), X2, X3, X4) → U61¹(X1, X2, X3, X4)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1, x2, x3, x4)  =  U61¹(x1, x3)
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U61^1₂

Status:

active₁: [1]
U61^1₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(184) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(185) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(186) TRUE

(187) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(X1, mark(X2)) → U52¹(X1, X2)
U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(X1, mark(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(189) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(mark(X1), X2) → U52¹(X1, X2)
U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(mark(X1), X2) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  U52¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U52^1₂

Status:

U52^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(191) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(active(X1), X2) → U52¹(X1, X2)
U52¹(X1, active(X2)) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(X1, active(X2)) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(active(X1), X2) → U52¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(194) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X1), X2) → U52¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(195) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(196) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(197) TRUE

(198) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(201) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U51^1₂

Status:

U51^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(202) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(203) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(204) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(active(X1), X2) → U51¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(205) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(active(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(206) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(207) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(208) TRUE

(209) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)
AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(210) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, mark(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(211) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(212) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(mark(X1), X2) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  AFTERNTH(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > AFTERNTH₂

Status:

mark₁: [1]
AFTERNTH₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(213) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)
AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(214) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(X1, active(X2)) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(215) Obligation:

Q DP problem:
The TRS P consists of the following rules:

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(216) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

AFTERNTH(active(X1), X2) → AFTERNTH(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
AFTERNTH(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(217) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(218) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(219) TRUE

(220) Obligation:

Q DP problem:
The TRS P consists of the following rules:

HEAD(active(X)) → HEAD(X)
HEAD(mark(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(221) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(mark(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1)  =  HEAD(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > HEAD₁

Status:

mark₁: [1]
HEAD₁: [1]

The following usable rules [FROCOS05] were oriented: none

(222) Obligation:

Q DP problem:
The TRS P consists of the following rules:

HEAD(active(X)) → HEAD(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(223) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

HEAD(active(X)) → HEAD(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
HEAD(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(224) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(225) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(226) TRUE

(227) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(228) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(229) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(230) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(231) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(232) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(233) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U42^1₂

Status:

active₁: [1]
U42^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(236) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U42^1₂

Status:

U42^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(237) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(238) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(239) TRUE

(240) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(241) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U41^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(243) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U41^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(244) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(245) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(246) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(247) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U41^1₂

Status:

active₁: [1]
U41^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(248) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(249) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U41^1₂

Status:

U41^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(250) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(251) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(252) TRUE

(253) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(254) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, mark(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(255) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(256) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U32^1₂

Status:

U32^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(257) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(258) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, active(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(259) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(active(X1), X2) → U32¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(260) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(261) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(262) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(263) TRUE

(264) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(265) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(266) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U31^1₂

Status:

U31^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, active(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(270) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(active(X1), X2) → U31¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(271) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(active(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(272) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(273) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(274) TRUE

(275) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(277) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U22^1₂

Status:

U22^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(279) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(280) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, active(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(281) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(active(X1), X2) → U22¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(282) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(active(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(283) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(284) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(285) TRUE

(286) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(287) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(288) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(289) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U21^1₂

Status:

U21^1₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(290) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(291) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, active(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(292) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(active(X1), X2) → U21¹(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(293) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(active(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(294) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(295) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(296) TRUE

(297) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)
SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(298) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(X1, mark(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(299) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)
SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(mark(X1), X2) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  SPLITAT(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SPLITAT₂

Status:

SPLITAT₂: [1,2]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(301) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)
SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(302) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(X1, active(X2)) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  x2
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(303) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(304) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SPLITAT(active(X1), X2) → SPLITAT(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(305) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(306) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(307) TRUE

(308) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SND(active(X)) → SND(X)
SND(mark(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(309) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(mark(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SND(x1)  =  SND(x1)
active(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > SND₁

Status:

mark₁: [1]
SND₁: [1]

The following usable rules [FROCOS05] were oriented: none

(310) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SND(active(X)) → SND(X)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(311) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SND(active(X)) → SND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SND(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]

The following usable rules [FROCOS05] were oriented: none

(312) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(313) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(314) TRUE

(315) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)
U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(316) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U12^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(317) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(318) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, active(X2), X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U12^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(319) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(320) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, X2, mark(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(321) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(322) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(active(X1), X2, X3) → U12¹(X1, X2, X3)
U12¹(X1, X2, active(X3)) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U12^1₂

Status:

active₁: [1]
U12^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(323) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(324) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2, X3) → U12¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2, x3)  =  U12¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U12^1₂

Status:

U12^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(325) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(326) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(327) TRUE

(328) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)
U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(329) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2), X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

U11^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(330) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(331) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, active(X2), X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x2)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

active₁: [1]
U11^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(332) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(333) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, X2, mark(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1

Lexicographic path order with status [LPO].
Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(334) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(335) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(active(X1), X2, X3) → U11¹(X1, X2, X3)
U11¹(X1, X2, active(X3)) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x1, x3)
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic path order with status [LPO].
Precedence:

active₁ > U11^1₂

Status:

active₁: [1]
U11^1₂: [1,2]

The following usable rules [FROCOS05] were oriented: none

(336) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(337) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2, X3) → U11¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2, x3)  =  U11¹(x1, x3)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Precedence:

mark₁ > U11^1₂

Status:

U11^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(338) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(339) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(340) TRUE

(341) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2, X3)) → ACTIVE(U11(mark(X1), X2, X3))
ACTIVE(U11(tt, N, XS)) → MARK(U12(tt, N, XS))
MARK(U11(X1, X2, X3)) → MARK(X1)
MARK(U12(X1, X2, X3)) → ACTIVE(U12(mark(X1), X2, X3))
ACTIVE(U12(tt, N, XS)) → MARK(snd(splitAt(N, XS)))
MARK(U12(X1, X2, X3)) → MARK(X1)
MARK(snd(X)) → ACTIVE(snd(mark(X)))
ACTIVE(U21(tt, X)) → MARK(U22(tt, X))
MARK(snd(X)) → MARK(X)
MARK(splitAt(X1, X2)) → ACTIVE(splitAt(mark(X1), mark(X2)))
ACTIVE(U22(tt, X)) → MARK(X)
MARK(splitAt(X1, X2)) → MARK(X1)
MARK(splitAt(X1, X2)) → MARK(X2)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ACTIVE(U31(tt, N)) → MARK(U32(tt, N))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
ACTIVE(U32(tt, N)) → MARK(N)
MARK(U22(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
ACTIVE(U41(tt, N, XS)) → MARK(U42(tt, N, XS))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
ACTIVE(U42(tt, N, XS)) → MARK(head(afterNth(N, XS)))
MARK(U32(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
ACTIVE(U51(tt, Y)) → MARK(U52(tt, Y))
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
ACTIVE(U52(tt, Y)) → MARK(Y)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(head(X)) → ACTIVE(head(mark(X)))
ACTIVE(U61(tt, N, X, XS)) → MARK(U62(tt, N, X, XS))
MARK(head(X)) → MARK(X)
MARK(afterNth(X1, X2)) → ACTIVE(afterNth(mark(X1), mark(X2)))
ACTIVE(U62(tt, N, X, XS)) → MARK(U63(tt, N, X, XS))
MARK(afterNth(X1, X2)) → MARK(X1)
MARK(afterNth(X1, X2)) → MARK(X2)
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
ACTIVE(U63(tt, N, X, XS)) → MARK(U64(splitAt(N, XS), X))
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(U64(pair(YS, ZS), X)) → MARK(pair(cons(X, YS), ZS))
MARK(U52(X1, X2)) → MARK(X1)
MARK(U61(X1, X2, X3, X4)) → ACTIVE(U61(mark(X1), X2, X3, X4))
ACTIVE(U71(tt, XS)) → MARK(U72(tt, XS))
MARK(U61(X1, X2, X3, X4)) → MARK(X1)
MARK(U62(X1, X2, X3, X4)) → ACTIVE(U62(mark(X1), X2, X3, X4))
ACTIVE(U72(tt, XS)) → MARK(XS)
MARK(U62(X1, X2, X3, X4)) → MARK(X1)
MARK(U63(X1, X2, X3, X4)) → ACTIVE(U63(mark(X1), X2, X3, X4))
ACTIVE(U81(tt, N, XS)) → MARK(U82(tt, N, XS))
MARK(U63(X1, X2, X3, X4)) → MARK(X1)
MARK(U64(X1, X2)) → ACTIVE(U64(mark(X1), X2))
ACTIVE(U82(tt, N, XS)) → MARK(fst(splitAt(N, XS)))
MARK(U64(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → ACTIVE(pair(mark(X1), mark(X2)))
ACTIVE(afterNth(N, XS)) → MARK(U11(tt, N, XS))
MARK(pair(X1, X2)) → MARK(X1)
MARK(pair(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fst(pair(X, Y))) → MARK(U21(tt, X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U71(X1, X2)) → ACTIVE(U71(mark(X1), X2))
ACTIVE(head(cons(N, XS))) → MARK(U31(tt, N))
MARK(U71(X1, X2)) → MARK(X1)
MARK(U72(X1, X2)) → ACTIVE(U72(mark(X1), X2))
ACTIVE(natsFrom(N)) → MARK(cons(N, natsFrom(s(N))))
MARK(U72(X1, X2)) → MARK(X1)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
ACTIVE(sel(N, XS)) → MARK(U41(tt, N, XS))
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
ACTIVE(snd(pair(X, Y))) → MARK(U51(tt, Y))
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(fst(X)) → ACTIVE(fst(mark(X)))
ACTIVE(splitAt(0, XS)) → MARK(pair(nil, XS))
MARK(fst(X)) → MARK(X)
MARK(natsFrom(X)) → ACTIVE(natsFrom(mark(X)))
ACTIVE(splitAt(s(N), cons(X, XS))) → MARK(U61(tt, N, X, XS))
MARK(natsFrom(X)) → MARK(X)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(tail(cons(N, XS))) → MARK(U71(tt, XS))
MARK(s(X)) → MARK(X)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(take(N, XS)) → MARK(U81(tt, N, XS))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(tail(X)) → MARK(X)
MARK(take(X1, X2)) → ACTIVE(take(mark(X1), mark(X2)))
MARK(take(X1, X2)) → MARK(X1)
MARK(take(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(snd(X)) → active(snd(mark(X)))
mark(splitAt(X1, X2)) → active(splitAt(mark(X1), mark(X2)))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(head(X)) → active(head(mark(X)))
mark(afterNth(X1, X2)) → active(afterNth(mark(X1), mark(X2)))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U61(X1, X2, X3, X4)) → active(U61(mark(X1), X2, X3, X4))
mark(U62(X1, X2, X3, X4)) → active(U62(mark(X1), X2, X3, X4))
mark(U63(X1, X2, X3, X4)) → active(U63(mark(X1), X2, X3, X4))
mark(U64(X1, X2)) → active(U64(mark(X1), X2))
mark(pair(X1, X2)) → active(pair(mark(X1), mark(X2)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U71(X1, X2)) → active(U71(mark(X1), X2))
mark(U72(X1, X2)) → active(U72(mark(X1), X2))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(fst(X)) → active(fst(mark(X)))
mark(natsFrom(X)) → active(natsFrom(mark(X)))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(nil) → active(nil)
mark(tail(X)) → active(tail(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
U11(mark(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, mark(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, mark(X3)) → U11(X1, X2, X3)
U11(active(X1), X2, X3) → U11(X1, X2, X3)
U11(X1, active(X2), X3) → U11(X1, X2, X3)
U11(X1, X2, active(X3)) → U11(X1, X2, X3)
U12(mark(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(X1, X2, X3)
snd(mark(X)) → snd(X)
snd(active(X)) → snd(X)
splitAt(mark(X1), X2) → splitAt(X1, X2)
splitAt(X1, mark(X2)) → splitAt(X1, X2)
splitAt(active(X1), X2) → splitAt(X1, X2)
splitAt(X1, active(X2)) → splitAt(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
head(mark(X)) → head(X)
head(active(X)) → head(X)
afterNth(mark(X1), X2) → afterNth(X1, X2)
afterNth(X1, mark(X2)) → afterNth(X1, X2)
afterNth(active(X1), X2) → afterNth(X1, X2)
afterNth(X1, active(X2)) → afterNth(X1, X2)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U61(mark(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, mark(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, mark(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, mark(X4)) → U61(X1, X2, X3, X4)
U61(active(X1), X2, X3, X4) → U61(X1, X2, X3, X4)
U61(X1, active(X2), X3, X4) → U61(X1, X2, X3, X4)
U61(X1, X2, active(X3), X4) → U61(X1, X2, X3, X4)
U61(X1, X2, X3, active(X4)) → U61(X1, X2, X3, X4)
U62(mark(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, mark(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, mark(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, mark(X4)) → U62(X1, X2, X3, X4)
U62(active(X1), X2, X3, X4) → U62(X1, X2, X3, X4)
U62(X1, active(X2), X3, X4) → U62(X1, X2, X3, X4)
U62(X1, X2, active(X3), X4) → U62(X1, X2, X3, X4)
U62(X1, X2, X3, active(X4)) → U62(X1, X2, X3, X4)
U63(mark(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, mark(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, mark(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, mark(X4)) → U63(X1, X2, X3, X4)
U63(active(X1), X2, X3, X4) → U63(X1, X2, X3, X4)
U63(X1, active(X2), X3, X4) → U63(X1, X2, X3, X4)
U63(X1, X2, active(X3), X4) → U63(X1, X2, X3, X4)
U63(X1, X2, X3, active(X4)) → U63(X1, X2, X3, X4)
U64(mark(X1), X2) → U64(X1, X2)
U64(X1, mark(X2)) → U64(X1, X2)
U64(active(X1), X2) → U64(X1, X2)
U64(X1, active(X2)) → U64(X1, X2)
pair(mark(X1), X2) → pair(X1, X2)
pair(X1, mark(X2)) → pair(X1, X2)
pair(active(X1), X2) → pair(X1, X2)
pair(X1, active(X2)) → pair(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U71(mark(X1), X2) → U71(X1, X2)
U71(X1, mark(X2)) → U71(X1, X2)
U71(active(X1), X2) → U71(X1, X2)
U71(X1, active(X2)) → U71(X1, X2)
U72(mark(X1), X2) → U72(X1, X2)
U72(X1, mark(X2)) → U72(X1, X2)
U72(active(X1), X2) → U72(X1, X2)
U72(X1, active(X2)) → U72(X1, X2)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
fst(mark(X)) → fst(X)
fst(active(X)) → fst(X)
natsFrom(mark(X)) → natsFrom(X)
natsFrom(active(X)) → natsFrom(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
take(mark(X1), X2) → take(X1, X2)
take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)
take(X1, active(X2)) → take(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.